Lattice Boltzmann method for Oldroyd-B fluids

• The Fokker–Planck equation of the dumbbell distribution is solved on a lattice.
• A new transport model for the dumbbells distribution functions is proposed.
• A 4 to 1 planar contraction is simulated and a lip vortex is observed.

A lattice method for modeling a solution of dumbbells in a Newtonian fluid, introduced by Onishi et al. (2005) [1], is implemented within an existing lattice Boltzmann framework (Körner et al. (2005)) [2]. The numerical scheme is compared with the analytical solution of steady state and start up simple shear flow and an oscillating velocity field under uniform motion. Furthermore, a new transport model is introduced and validated for an oscillating velocity field under uniform motion. Both tests are in good agreement with the analytical solution. Additionally, a 4 to 1 planar contraction is simulated to test if the method is capable of reproducing complex viscoelastic flows. A lip vortex at the entrance corner is observed. A second benchmark is the four-roller setup. The overall results are in good agreement with previous studies.